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Under certain circumstances, light can propagate with a phase velocity or group velocity greater than c, the vacuum velocity of light.
At a first glance, this may suggest the possibility of superluminal transmission of information – in striking contradiction with Einstein's theory of relativity, which shows that superluminal transmission would be associated with a violation of causality.
However, careful inspection of the accepted physical theories (in particular, of Maxwell's theory of electromagnetism, Einstein's theory of relativity, and quantum electrodynamics) shows that these definitely do not allow any non-causal effects.
Any attempt to demonstrate such effects on the basis of these theories must fail for logical reasons and is thus based on fundamental misconceptions, although the flaw may not always be easy to identify.

Two relatively simple examples of apparently (but not really) non-causal transmission are given here:

The phase velocity of light can be higher than c, e.g. for X-rays or near optical resonances of atoms.
Even a simple mechanical model (e.g. a sequence of coupled pendulums) can exhibit a phase velocity above c.
However, it is not possible to attach a real signal to such a phase front; signal transmission requires some kind of modulation.

Information can be transmitted by sending (or not sending) pulses.
The information then appears to propagate with the group velocity, i.e. the velocity of the peaks.
Group velocities larger than c then appear to lead to superluminal transmission and can occur e.g. in tunneling experiments.
However, the error lies in identifying the peak of the pulse with the temporal position of the carried information [1].
For example, a Gaussian-shaped pulse can be detected long before its peak due to the rise of intensity at earlier times.
Therefore, a different kind of signal must be considered, where no information at all is sent out before a certain moment of time.
For such signals, it can be proven that the earliest time at which that switching event can be observed is limited exactly by propagation with the vacuum velocity c [1].
A so-called precursor is traveling with that speed, but is normally too weak to be detected, except in certain circumstances [10].

Of course, it is conceivable that the currently accepted physical theories, e.g. Einstein's theory of relativity, will have to be revised or replaced with newer theories, possibly even allowing for superluminal propagation.
Such theories exist already, but so far there is no convincing evidence that could support the withdrawal of relativity in the favor of an alternative theory.